There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Physical Review B

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Emergent Nonlinear Phenomena in Bose-Einstein Condensates

Basic Mean-Field Theory for Bose-Einstein Condensates.- Basic Mean-Field Theory for Bose-Einstein Condensates.- Bright Solitons in Bose-Einstein Condensates.- Bright Solitons in Bose-Einstein Condensates: Theory.- Bright Solitons: Summary of Experimental Techniques.- Dark Solitons in Bose-Einstein Condensates.- Dark Solitons in Bose-Einstein Condensates: Theory.- Dark Solitons in BECs: The First Experiments.- Nonlinear Localization of BECs in Optical Lattices.- Nonlinear Localization of BECs in Optical Lattices.- Multi-Dimensional Solitons in Bose-Einstein Condensates.- Multidimensional Solitons: Theory.- Experiments on Multidimensional Solitons.- Vortices in Bose-Einstein Condensates.- Vortices in Bose-Einstein Condensates: Theory.- Vortices in Bose-Einstein Condensates: Experiments.- Vortex Lattices.- Vortex Lattices in Bose-Einstein Condensates: Theory.- Vortex Lattices in Bose-Einstein Condensates: Experiments.- Optical Lattices.- Optical Lattices: Theory.- Bose-Einstein Condensates in Optical Lattices: Experiments.- Multi-Component Bose-Einstein Condensates.- Multi-Component Bose-Einstein Condensates: Theory.- Multi-Component Condensates: Experiment.- Manipulations of Coherent Matter-Waves.- Manipulations of Coherent Matter-Waves.- Beyond Gross-Pitaevskii Mean Field Theory.- Beyond Gross-Pitaevskii Mean-Field Theory.- Asymptotic Reductions of the Gross-Pitaevskii Equation.- Asymptotic Reductions of the Gross-Pitaevskii Equation.

Emergent nonlinear phenomena in Bose-Einstein condensates : theory and experiment

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

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Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

We revisit a classic study [D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998)] of interpenetrating Bose-Einstein condensates in the hyperfine states $|F=1,{m}_{f}=\ensuremath{-}1⟩\ensuremath{\equiv}|1⟩$ and $|F=2,{m}_{f}=+1⟩\ensuremath{\equiv}|2⟩$ of $^{87}\mathrm{Rb}$ and observe striking new nonequilibrium component separation dynamics in the form of oscillating ringlike structures. The process of component separation is not significantly damped, a finding that also contrasts sharply with earlier experimental work, allowing a clean first look at a collective excitation of a binary superfluid. We further demonstrate extraordinary quantitative agreement between theoretical and experimental results using a multicomponent mean-field model with key additional features: the inclusion of atomic losses and the careful characterization of trap potentials (at the level of a fraction of a percent).

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Nonequilibrium Dynamics and Superfluid Ring Excitations in Binary Bose-Einstein Condensates

We consider nonlinear analogs of parity-time- ($\mathcal{PT}$-) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and odd excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter $\ensuremath{\varepsilon}$ controlling the strength of the $\mathcal{PT}$-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as $\ensuremath{\varepsilon}$ is further increased, the ground state and first excited state, as well as branches of higher multisoliton (multivortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear $\mathcal{PT}$ phase transition---thus termed the nonlinear $\mathcal{PT}$ phase transition. Past this critical point, initialization of, e.g., the former ground state, leads to spontaneously emerging solitons and vortices.

https://works.bepress.com/panos_kevrekidis/215/download/

Ricardo Carretero-González

Papers

Emergent Nonlinear Phenomena in Bose-Einstein Condensates

Emergent nonlinear phenomena in Bose-Einstein condensates : theory and experiment

Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

Nonequilibrium Dynamics and Superfluid Ring Excitations in Binary Bose-Einstein Condensates

Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions